Chemical Principles: The Quest for Insight
by Peter Atkins, Loretta Jones, Leroy Laverman
Atkins' Chemical Principles (7th edition, but 5th or 6th are also good) is pretty much a one-stop shop for the introductory theory you need to understand chemistry. It requires basic knowledge of integral calculus, although equations involving calculus could be skipped for readers less enthusiastic about math. Atkins' book also offers a very basic intro to chemistry for students who may not have any introduction. The rest of the book progresses through general chemistry in a series of 11 chapters involving an intro to quantum mechanics, molecular theory, materials, thermodynamics, equilibrium, acid-base & redox, kinetics, the elements, coordination chemistry, nuclear chemistry, followed by intro orgo- and biochem. There are examples, as well as problems and exercises at the end of chapters/sections (odd answers in back), more than enough to reinforce concepts. Another feature in the book is that is has various "boxes" that talk about cool applications of chemistry and the principles you've just learned. Atkins' Chemical Principles is more detailed than Zumdahl's Chemistry, another book commonly read by beginner chemistry olympians. However, it isn't much harder to understand and will also introduce you to more advanced concepts.
Find more information about cost and availability at Open Library.